In this paper, the authors investigate the achievable rate and ergodic capacity of a general non-symmetric half-duplex Non-orthogonal Amplify-and-Forward (NAF) single-relay channel in Rayleigh fading environments, assuming that the channel state information is only available at the destination. The considered channel, which captures path-loss and shadowing effects over the transmission links, includes the symmetric one as a special case. At first, for a given power allocation scheme, simple and closed-form expressions of the upper and lower bounds on the achievable rate are derived.